Table of Contents
International Journal of Combinatorics
Volume 2012 (2012), Article ID 406250, 11 pages
http://dx.doi.org/10.1155/2012/406250
Research Article

Variations of the Game 3-Euclid

Department of Mathematics, La Trobe University, Melbourne, VIC 3086, Australia

Received 30 November 2011; Revised 27 December 2011; Accepted 5 January 2012

Academic Editor: Toufik Mansour

Copyright © 2012 Nhan Bao Ho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present two variations of the game 3-Euclid. The games involve a triplet of positive integers. Two players move alternately. In the first game, each move is to subtract a positive integer multiple of the smallest integer from one of the other integers as long as the result remains positive. In the second game, each move is to subtract a positive integer multiple of the smallest integer from the largest integer as long as the result remains positive. The player who makes the last move wins. We show that the two games have the same 𝒫 -positions and positions of Sprague-Grundy value 1. We present three theorems on the periodicity of 𝒫 -positions and positions of Sprague-Grundy value 1. We also obtain a theorem on the partition of Sprague-Grundy values for each game. In addition, we examine the misère versions of the two games and show that the Sprague-Grundy functions of each game and its misère version differ slightly.